Write an equation in point-slope form for the line that passes through the points \((-4, 6)\) and \((-2, 22)\).
Calculate the slope \(m\).
Using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), we find \(m = \frac{22 - 6}{-2 - (-4)} = \frac{16}{2} = 8\).
Write the point-slope form equation.
Using the point-slope form \(y - y_1 = m(x - x_1)\), we have \(y - 6 = 8(x - (-4))\).
The equation is \(\boxed{y - 6 = 8(x + 4)}\).
Write an equation in point-slope form for the line that passes through the points \((1, -3)\) and \((4, -15)\).
Calculate the slope \(m\).
Using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), we find \(m = \frac{-15 - (-3)}{4 - 1} = \frac{-12}{3} = -4\).
Write the point-slope form equation.
Using the point-slope form \(y - y_1 = m(x - x_1)\), we have \(y + 3 = -4(x - 1)\).
The equation is \(\boxed{y + 3 = -4(x - 1)}\).
Write an equation in point-slope form for the line that passes through the points \((4, -6)\) and \((6, -4)\).
Calculate the slope \(m\).
Using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), we find \(m = \frac{-4 - (-6)}{6 - 4} = \frac{2}{2} = 1\).
Write the point-slope form equation.
Using the point-slope form \(y - y_1 = m(x - x_1)\), we have \(y + 6 = 1(x - 4)\).
The equation is \(\boxed{y + 6 = 1(x - 4)}\).
Write an equation in point-slope form for the line that passes through the points \((3, 3)\) and \((6, 7)\).
Calculate the slope \(m\).
Using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), we find \(m = \frac{7 - 3}{6 - 3} = \frac{4}{3}\).
Write the point-slope form equation.
Using the point-slope form \(y - y_1 = m(x - x_1)\), we have \(y - 3 = \frac{4}{3}(x - 3)\).
The equation is \(\boxed{y - 3 = \frac{4}{3}(x - 3)}\).
Convert the equation \(y - 1 = \frac{4}{5}(x + 5)\) to slope-intercept form.
Identify \(y_1\), \(m\), and \(x_1\).
Here, \(y_1 = 1\), \(m = \frac{4}{5}\), and \(x_1 = -5\).
Convert to slope-intercept form.
Solving for \(y\), we get \(y = \frac{4}{5}x + 5\).
The equation is \(\boxed{y = \frac{4}{5}x + 5}\).
Convert the equation \(y + 5 = -6(x + 7)\) to slope-intercept form.
Identify \(y_1\), \(m\), and \(x_1\).
Here, \(y_1 = -5\), \(m = -6\), and \(x_1 = -7\).
Convert to slope-intercept form.
Solving for \(y\), we get \(y = -6x - 47\).
The equation is \(\boxed{y = -6x - 47}\).
Convert the equation \(y + 6 = -\frac{3}{4}(x + 8)\) to slope-intercept form.
Identify \(y_1\), \(m\), and \(x_1\).
Here, \(y_1 = -6\), \(m = -\frac{3}{4}\), and \(x_1 = -8\).
Convert to slope-intercept form.
Solving for \(y\), we get \(y = -\frac{3}{4}x - 12\).
The equation is \(\boxed{y = -\frac{3}{4}x - 12}\).
Convert the equation \(y + 2 = \frac{1}{6}(x - 4)\) to slope-intercept form.
Identify \(y_1\), \(m\), and \(x_1\).
Here, \(y_1 = -2\), \(m = \frac{1}{6}\), and \(x_1 = 4\).
Convert to slope-intercept form.
Solving for \(y\), we get \(y = \frac{1}{6}x - \frac{2}{3}\).
The equation is \(\boxed{y = \frac{1}{6}x - \frac{2}{3}}\).
The equations in point-slope form are:
\(\boxed{y - 6 = 8(x + 4)}\)
\(\boxed{y + 3 = -4(x - 1)}\)
\(\boxed{y + 6 = 1(x - 4)}\)
\(\boxed{y - 3 = \frac{4}{3}(x - 3)}\)
The equations in slope-intercept form are:
\(\boxed{y = \frac{4}{5}x + 5}\)
\(\boxed{y = -6x - 47}\)
\(\boxed{y = -\frac{3}{4}x - 12}\)
\(\boxed{y = \frac{1}{6}x - \frac{2}{3}}\)
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